Towards the beginning of a course on knowledge (we were reading the beginning of Theaetetus) we discussed definitions and what a definition or rather a good definition is. I'd like to know more about definability. Is there a particular word/concept/notion that has been undefinable? I know that every language must have undefinable/primitive notions/concepts* but aside from within formal systems in logic, set theory, and mathematics have I seen one. Has anyone been able to prove that a particular word is undefinable? If so, how? References are appreciated too, thanks.
*To prove the existence of primitive notions, we notice that a good definition of a word or concept does not use itself. For example, a definition of orange juice as juice that comes from oranges is not a good one (of course orange juice is a juice, but what is a juice, namely orange juice?). Moreover, if we start to exhaustively define every notion in terms of other notions (assuming the class of words or concepts/notions are countable), surely there must be a word (towards the end of our list) in which we cannot define in terms of another notion (maybe because this notion precedes all other notions, i.e. every other notion is defined, possibly indirectly, in terms of this final notion). Defining this (last) concept in terms of another previously defined will make them circular. Therefore, there must exist a primitive notion undefinable in terms of others and unsatisfactorily defined if in terms of itself.
Thanks in advance!!
The following comment clarified the sort of circular reasoning I refer to above:
I think this is the problem the OP has in mind: dictionary.reference.com/browse/hill: "a natural elevation of the earth's surface small than that of a mountain." dictionary.reference.come/browse/mountain: "a natural elevation of the earth's surface rising more or less abruptly to a summit, and attaining an altitude greater than that of a hill. - DBK
Here the dictionary defines a hill in terms of a mountain, and mountain in terms of a hill! This type of defining is circular... we understand the definitions because of our experience with hills and mountains. If hills and mountains were completely foreign to us, these definitions wouldn't make much sense (try explaining this to a child without using pictures). This type of phenomenon will must occur if we try to define all concepts exaustively. This shows that there are notions that we must take as primitive or undefinable (understanding a concepts is not the same as defining it here - to see this, we understand the conepts of good, just, knowldge, etc. and can certainly give examples of each, but defining it is much more difficult).
Many concepts have been analysed by Plato through Socrates' dialogues. Some of these include piety, justice, knowledge,... At the end of each dialogue, they conclude with what the concept being analysed isn't! I'm wondering whether any of these concepts (or any other) have been shown to be undefinable.